Lesson 25: Ch.
30 (1) – Faraday’s and Lenz’s Law
1 Induction 1. dB
dt EMF if B changes in time (like
EMP)
In the early 19th Century, Michael Fara- dA
day discovered that a changing magnetic 2. dt EMF if A changes in time (like
field could induce an emf (and hence a a rail gun)
current) in a loop. Specifically, he found dθ
3. dt EMF with rotation (like a gen-
that a current was generated when the erator)
magnetic flux, ΦB changed. This is now
known as Faraday’s Law:
−dΦB
~ ·A
ΦB = B ~ = BA cos θ 4 Lenz’s Law
E= dt
1. B is the magnetic field in Tesla. Remember that negative sign in Fara-
If we were “vector ninjas” that minus
day’s Law? If we are careful about how
sign would help give a direction. The
2. A is the area of the loop. the dot product is taken it would tell us
derivative term describes doing some-
the direction of the current in the loop.
thing with the magnet to change the 3. θ is the angle between the two vec- However, Lenz’s Law is a good way to
“Magnetic Flux” in time. Like any B tors. know the answer without math. It tells
field effect, N loops multiplies the effect
us the direction of the induced current.
N times. Magnetic flux is measured in a unit
−N dΦB called a weber (Wb), 1 Wb = 1 T m2 The induced current is in the direction
E= dt such that the field produced by the induced
We’ve known for ages that you rub current opposes the change in flux which
two sticks together to make fire – here 3 How does Flux change produced it.
Faraday’s Law is telling you what you in Time? Steps to using Lenz’s law:
have to do to a coil of wire and a magnet
to make electricity. The chain rule is used, considering B, A, θ 1. Which way is the existing B field?
depending on the problem might be func-
2. Is the flux increasing or decreasing?
tions of time.
2 Magnetic Flux d(BA cos θ) 3. If decreasing, induced current
E = −N dΦ
dt = −N
B
dt
moves to make B field in the same
Magnetic flux is just like our earlier def-
R
~ · dA
~ E = −N (A cos θ dB dA
dt + B cos θ dt − direction as the existing field.
inition of Electric flux: ΦB = B
BA sin θ dθ
dt )
For only the special case when B is uni- 4. If increasing, induced current moves
form throughout the area of the loop, this The trick to each problem is identify- to make B field in the opposite di-
integral simplifies to this ing which variables are changing! rection from the existing field.
1
Exercise 1: A circular wire loop of
radius 1.0 cm is oriented so that its nor-
mal makes a 60◦ angle with a uniform
1.0 mT magnetic field. (a) What is the
magnetic flux through the loop? (b) If the
loop is rotated so that its normal makes
a 45◦ angle with the magnetic field, does Exercise 2: Suppose that when the
the flux increase, decrease, or remain the switch is closed, the current in the right- Exercise 3: What is the direction
same? hand loop takes 1.0 µs to build to its (left, right, or zero) of the current
steady-state value, and that during this through the ammeter in each case?
time, the magnetic flux through the left-
hand loop increases steadily from zero to (a) The magnet’s north pole is moved
1.0 × 10−8 Wb. toward the loop as shown.
(a) While the flux is increasing, what (b) The magnet is held still while its
are the magnitude and direction of north pole is inside the loop.
the induced emf in the left-hand (c) The magnet is moved away from the
loop? loop.
(b) During this time, what is the direc- (d) The magnet is turned around and
tion of the current through the am- moved toward the loop, south end
meter? first.
(c) Once the current through the (e) The magnet is held still with the
right-hand loop stabilizes, the flux south end in the loop.
through the loops remains constant. (f) With its south pole in the loop,
Now how much emf is induced in the the magnet is moved away from the
left-hand loop? loop.
(d) When the switch is opened, the cur-
rent in the right-hand loop decreases
steadily to zero in 1.0 × 10−7 s. Dur-
ing this time, what are the magni-
tude and direction of the induced
emf in the left-hand loop?
(e) After the current is stopped, what is
the emf in the left-hand loop?
